= Solution
For an isolated approximately spherical <mass> distribution outside its luminous extent, retain the gravitational <force> $GMm/r^2$ and use the specified low-acceleration inertial law. <circular motion> then gives
$$
\frac{GMm}{r^2}=\frac m{a_0}\left(\frac{v^2}{r}\right)^2,
\qquad\boxed{v^4=GMa_0}.
$$
The <radius> cancels when enclosed luminous <mass> is effectively constant. This is the asymptotic <modified Newtonian dynamics> result associated with the <baryonic Tully-Fisher relation>; an arbitrary extended disk cannot everywhere be replaced exactly by a point <mass>.
At the disk edge,
$$
a(R)=\frac{(9\times10^4)^2}{3\times10^{20}}
=\boxed{2.7\times10^{-11}\,\mathrm{m\,s^{-2}}=0.27a_0}.
$$
Using $G=6.67\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$ gives
$$
M_{\rm MOND}=\frac{v^4}{Ga_0}=9.84\times10^{39}\,\mathrm{kg}
=\boxed{4.9\times10^9M_\odot}.
$$
This is an order-of-magnitude isolated, low-acceleration estimate. The edge <acceleration> is below $a_0$, but not extremely small compared with it; a full interpolation law, disk geometry and an external gravitational field can affect a more precise <mass> fit.
For the <spherical mass profile for a flat rotation curve>, assume spherical symmetry where the halo dominates. A <flat rotation curve> gives $M(<r)=v^2r/G$, and differentiating $M(<r)=4\pi\int_0^r\rho(s)s^2\,ds$ yields
$$
\boxed{\rho(r)=\frac{v^2}{4\pi Gr^2}}.
$$
Here $\rho$ is the density of the spherical gravitating model; it is not an exact deprojection of a thin luminous disk or a proof that the central density must diverge. At the outermost stated measured <radius>,
$$
M(<2R)=\frac{v^2(2R)}G=7.29\times10^{40}\,\mathrm{kg}
=\boxed{3.6\times10^{10}M_\odot}.
$$
This gives the requested galaxy-plus-halo <mass> estimate if the model is truncated at the outermost observed <radius> $2R$. \b[The observation itself constrains enclosed <mass> within $2R$]; any additional outer halo remains unconstrained. Extending $\rho\propto r^{-2}$ to infinity would give divergent <mass>. Subtracting the MOND-inferred luminous <mass> to obtain a Newtonian halo <mass> would mix independent model-dependent estimates without an independently measured luminous <mass>.
In the <Bullet Cluster> collision, most observed baryonic <mass> is hot gas. <ram pressure> slows and <shocks> the gas, while the galaxies pass through much more freely. <weak gravitational lensing> maps the projected gravitational <mass> without requiring that the gas be in <hydrostatic equilibrium>. The dominant lensing peaks lie closer to the galaxies than to the displaced X-ray gas. This supports a dominant weakly collisional <dark matter> component under the usual gravity model. It is stronger evidence than a mass-to-light discrepancy alone because distinct components separate during the same encounter. The inference needs background-source shape/redshift <calibration>, projection control and uncertainty in the lensing reconstruction; it does not identify a <WIMP> species or by itself exclude every modified-gravity theory.<A direct empirical proof of the existence of dark matter by Douglas Clowe and others>[Abstract and comparison of lensing mass, optical galaxies and X-ray gas]{ref}
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2013/paper-1-bullet-components.png]
{title=Original conceptual merger sketch showing galaxies and dominant lensing mass displaced from slowed X-ray gas; illustrative positions, not observational data}
For <dark-matter direct detection>, consider <XENON100> at Gran Sasso. A halo <WIMP> can undergo <elastic scattering> from a <xenon> <atomic nucleus>, leaving a recoil of only a few to tens of keV. The expected rate is very low, so a large target, low threshold and long exposure must be combined with background rejection. The <nuclear-recoil kinematics of WIMP scattering> gives $E_{R,\max}=2\mu_{\chi N}^2v^2/m_N$ in the nonrelativistic limit, where $\mu_{\chi N}$ is the <reduced mass> and <masses> and <velocities> use consistent units. This explains why low-energy response <calibration> matters.
The dual-phase <xenon> detector measures prompt <particle scintillation> S1 and delayed <electroluminescence> S2 from <ionization> electrons extracted into gas. Their time separation gives interaction depth; the S2 light pattern supplies transverse position. An inner <fiducial volume> uses <xenon>'s self-shielding to suppress external backgrounds. The S2/S1 distribution statistically separates <nuclear recoils> from <electron recoils>; it is not perfect event-by-event identification. Underground siting reduces cosmic-ray backgrounds, while passive shielding, low-radioactivity components, <xenon> purification and a surrounding <active detector veto> reduce other events.<The XENON100 Dark Matter Experiment by the XENON100 Collaboration>[Sections 2–3, two-phase detection, detector design and background control]{ref}
Neutrons can also make <nuclear recoils>, so charge-to-light discrimination alone cannot eliminate them. Multiple-scatter rejection, shielding and background modelling remain necessary. Gamma-source and neutron-source <calibrations> map electron- and nuclear-recoil distributions, <energy> scale and selection acceptance. Position-dependent light collection and charge losses need correction; photon-counting fluctuations and uncertain low-energy yields limit threshold sensitivity. Blind event selection and independent sidebands guard against tuning cuts to apparent candidates. The 224.6-live-day, 34 kg search found two benchmark candidates against about one expected background event, consistent with background. Its interaction limit therefore depends on the assumed halo distribution, <WIMP> <mass> and nuclear response; it is not a detection or an exclusion of all <dark matter>.<Dark Matter Results from 225 Live Days of XENON100 Data by the XENON100 Collaboration>[Printed pp. 2–4, calibrations, blind selection and backgrounds; abstract for exposure and result]{ref}
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